TF 

222 


731 


UTATiCN 


RAIL  ;  OTHER  EARTHWORK 


(AT 


•I 


liliiil 


Illliii 


GIFT   OF 
Charles  1,1.   Titus 


UNIVERSITY  FARM 


1  i 


TABLES 


FOR  THE  COMPUTATION 


OF 


Railway  and  Other  Earthwork. 

V 


COMPUTED   BY 


C.  L.  CRANDALL,  C.  E., 

Associate  Professor  of  Civil  Engineering,  Cornell  University 
Member  of  American  Society  of  Civil  Engineers. 


SECOND  EDITION. 


NEW  YORK: 

JOHN  WII,EY  &  SONS, 

53  EAST  TENTH  STREET. 

1893- 


UN-IV  '*  CALIFORNIA 

LIBRARY 

BRANCH  OF  THE 


COPYRIGHT,  1893, 
BY  CHARTS  L.  CRANDAU,. 


ANDRUS   &   CHURCH, 

PRINTERS, 
ITHACA,    N.  Y. 


Preface  to  the  First  Edition. 


THE  object  of  these  tables  is  to  present  a  convenient  aid  in  the  Com- 
putation of  Railway  Earthwork.  The  volume  is  first  found  by  the 
approximate  method  of  averaging  end  areas,  to  which  can  be  added,  if 
desired,  a  correction  due  to  the  strict  prismoidal  formula,  or  a  correc- 
tion due  to  a  modification  of  it  for  irregular  ground.  This  separation 
reduces  the  labor,  if  both  parts  are  computed  at  once,  while  it  allows 
of  computing  only  the  approximate  value  at  first,  which  will  answer 
for  all  purposes  except  the  final  estimate,  leaving  the  prismoidal  cor- 
rection to  be  computed  at  leisure  on  those  volumes  only  where  it  is 
large  enough  to  be  appreciable. 

The  Table  for  "Volume  by  Averaging  End  Areas,"  is  a  table  for 
triangular  prisms,  and  it  will  give  the  approximate  volume  of  triangu- 
lar prismoids  in  cubic  yards  for  a  length  of  one  hundred  feet,  between 
the  parallel  triangular  ends,  by  taking  the  sum  of  the  two  numbers 
found  with  the  base  and  altitude  of  each  of  the  two  end  triangles  as 
arguments.  The  table  is  thus  independent  of  side  slope,  or  width  of 
roadbed,  and  will  apply  to  borrow  pits,  trenches,  etc.,  by  dividing  up 
the  cross  sections  into  triangles  and  trapezoids. 

The  method  of  dividing  up  the  sections  for  three-level,  five-level 
and  irregular  ground,  while  not  essential,  is  believed  to  give  the  least 
labor  in  computing,  either  with  or  without  tables. 

The  method  for  irregular  ground  is  due  to  J.  W.  Davis,  and  was  pub- 
lished in  his  Formula  for  R.  R.  Earthwork,  1876. 

The  Table  for  "Prismoidal  Correction"  is  a  table  for  triangular 
prismoids,  and  it  will  give  the  exact  correction  to  be  applied  to  the  ap- 
proximate volume  found  above  for  all  solids  which  can  be  divided  into 
triangular  prismoids,  having  plane  parallel  ends,  and  planes  or  hyper- 
bolic paraboloids  for  side  faces.  By  using  A.  M.  Wellington's  method 
for  irregular  ground,  that  for  the  purposes  of  the  prismoidal  correction 
only,  the  ground  may  be  called  regular,  the  correction  is  easily  and 
quickly  applied,  to  three-level,  five-level,  and  irregular  ground.  It 
may  be  noted  that  when  either  the  bases  or  altitudes  of  the  end  trian- 
gles are  equal,  the  correction  disappears. 

While  quantities  are  only  given  for  full  feet  the  corrections  for  tenths 
are  easily  taken  out  from  the  margin  without  multiplication  and  can 
be  added  mentally  with  but  slight  increase  of  labor. 

The  Tables  have  been  carefully  recomputed  in  print,  and  it  is  confi- 
dently believed  they  are  free  from  errors. 

PREFACE  TO  THE  SECOND  EDITION. 

The  first  edition  was  not  electrotyped,  and  advantage  has  been  taken 
of  this  to  entirely  recast  the  text,  adding  a  proof  of  the  prismoidal  for- 
mula and  a  comparison  with  others  in  use,  a  formula  for  correction  for 
curvature,  and  rules  for  cross  sectioning. 

ITHACA,  N.  Y.,  August,  1893. 


ZZ  76, 


Contents. 


DERIVATION  OF  PRISMOIDAL  FORMULA  AND  COMPARI- 
SON WITH  OTHERS  IN  USE. 
SEC.  PAGE. 

1.  Prisinoidal  Formula i 

2.  Averaging  End  Areas 2 

3.  Middle  Areas 3 

4.  Equivalent  Mean  Heights 3 

5.  Triangular  Prismoid  with  Curved  Center  Line 4 

VOLUME   FOR    RAILWAY   EARTHWORK    IN    CUBIC   YARDS    BY 
AVERAGING    END   AREAS. 

6.  Formula  for  Three-level  Ground 7 

7.  Five-level  Ground 7 

8.  Irregular  Ground 8 

9.  Tables 10 

10.  Example,  Three  level  Ground n 

11.  Example,   Five-level  Ground n 

12.  Example,   Irregular  Ground 12 

13.  Borrow  Pits 13 

14.  Sidehill  Work 13 

15.  End  Sections       .    .            14 

16.  Correction  for  Curvature 14 

17.  Example,  Correction  for  Curvature 16 

PRISMOIDAL   CORRECTION    IN    CUBIC    YARDS. 

18.  Three  level  Ground 17 

19.  Five-level  Ground 17 

20.  Irregular  Ground,  Borrow  Pits,  etc.  .       17 

21.  Table  of  Prisinoidal  Corrections 18 

22.  Example,  Three- level  Ground 18 

23.  Example,  Five-level  Ground 19 

24.  Example,  Irregular  Ground 19 

25.  Example,  Borrow  Pits 20 

COMPUTATION,  END  AREAS  WITH  PRISMOIDAL  CORRECTION. 

26.  Three-level  Ground 21 

27.  Irregular  Ground 21 

28.  Rules  for  cross-sectioning 23 

Table  of  Triangular  Prismoids  for  Railway  Earthwork,  in  cubic  yards, 

by  Averaging  End  Areas. 

Table  of  Prismoidal  Corrections  for  Triangular  Prismoids  for  Rail- 
way Earthwork. 


Derivation  of  Prismoidal  Formula  and 
Comparison  with  Others  in  Use. 


Section     xx> 
First  End' 


cU(d 


i.  Prismoidal  Formula. — In  computing  railway  earthwork,  it  is 
customary  to  measure  sections  perpendicular  to  the  center  line  at  sta- 
tions 100  feet  apart,  and  at  as  many 
intermediate  points  as  may  be  found 
necessary. 

To  find  the  volume  between  two 
such  parallel  sections  : 

Take  at  first  the  simple  case  shown 
in  Fig.  i,  where  the  section  is  straight 
from  the  center  to  the  slope  stake  on 
each  side,  and  where  straight  lines 
joining  corresponding  points  of  the 
upper  lines  of  the  sections  lie  in  the 
ground  surface.  The  ground  surface 
thus  formed  is  a  hyperbolic  parabo- 
loid, a  characteristic  property  of 
which  is  that  all  planes  parallel  to 
the  end  sections  cut  the  surface  in 
straight  lines,  whose  increments  for 
decrements)  in  length,  starting  from 
either  end,  are  proportional  to  their 
distances  from  that  end. 

With  the  notation  of  Fig.  i,  the 
area  of  the  right-hand  side  of  the  first 
section  (including  the  grade  triangle 
under  the  roadbed), 

A  =  $(a  +  c)d. 

At  the  distance  z,  since  all  the  linear  dimensions  of  the  section  can 
be  interpolated, 

•—  *.  ~1  •—  „  *1 

(a) 


Plan 


Section 
Second 


Fig  .. 


Elementary  volume  of  length  dz  =  area  into  length 


Integrating  between  the  limits  z  =o  to  z  =  /, 

Volume  =  C1AZ  dz 
J  o 


PRISMOIDAL  AND  OTHER  FORMULAS.  [|  2, 


-c)     +  (c'-c)  (d'-#) 


Volume  = 

26 


•-L  <*+0  /  5  ,  a+c  d'l     c'—c  d'l     a+cdl^      </—  cd_l 
2         6+     23"       2622+26 


Vol.  = 


where  Am  is  the  area  of  the  mid  section  found  by  making  £  —  /-?-  2 
in  (a). 

1.  e.,  the  true  volume  equals  the  sum  of  the  areas  of  the  end  sections 
and  four  times  that  of  the  mid  section,  multiplied  by  one-sixth  the 
perpendicular  length  between  sections. 

This  is  the  prismoidal  formula,  and  the  first  proof  of  its  application 
to  an  earthwork  solid,  as  above,  is  due  to  Professor  Gillespie.* 

It  is  evident  that  an  earthwork  solid  with  polygonal  end  sections 
lying  in  parallel  planes  can  be  divided  into  solids  with  triangular  bases 
as  above,  when  there  are  the  same  number  of  sides  in  the  end  poly- 
gons and  lines  joining  corresponding  points  of  these  sections  lie  in  the 
bounding  surfaces  ;  therefore  the  prismoidal  formula  will  give  the  true 
volume  for  all  such  solids.  The  solid  thus  described  will  be  called  a 
prismoid  ;  i.  e.,  a  prismoid  will  be  defined  as  a  solid  having  two  plane 
polygonal  parallel  bases  with  the  same  number  of  sides  in  each,  joined 
by  either  planes  or  hyperbolic  paraboloids  for  side  faces.  With  trian- 
gles for  bases,  the  prismoid  becomes  a  triangular  prismoid. 

2.  Averaging  End  Areas.  —  This  is  a  method  in  common  use  on 
account  of  its  simplicity.     Applying  it  to  the  triangular  prismoid  of 
$  i,  and  including  the  grade  triangle  for  simplicity, 

By  averaging  end  areas, 


2  2 

True  volume  by  the  prismoidal  formula, 


Vol   -          d       >d>\  2l  \c+c'  d+d'  2/ 

>p.2         6^2          6^2       2       34         2       3 

Subtracting  Vol.e  from  the  true  value,  Vol.p, 

Vol.  -Vol    =-^di-^d>L  +  Wal_ 

23  2  3*23 

*  See  Roads  and  Railroads,  New  York,  1847. 


Bq-3] 


EQUIVALENT  MEAN  HEIGHTS. 


(2) 


where  Cor.e  is  the  correction  to  be  applied  to  the  volume  found  by  aver- 
aging end  areas  in  order  to  obtain  the  true  volume. 

Usually  in  passing  from  lighter  to  heavier  work  c'  >  c  and  dr  >  d 
(or  conversely  from  heavier  to  lighter),  making  the  above  correction 
usually  negative;  i.  <?.,  the  method  by  averaging  end  areas  gives  on 
the  average  too  large  a  volume. 

3.  Middle  Areas. — The  linear  dimensions  of  the  mid  section  will 
be  the  means  of  those  of  the  end  sections,  giving 


Subtracting  this  from  the  true  value, 


Vol.p-Vol.m=  - 
2 


___  _(d+d/) 

3       24         24  82  3 


24 


(3) 


where  Cor.m  is  the  correction  to  be  applied  to  the  volume  found  by  mid- 
dle areas  in  order  to  obtain  the  true  volume. 

This  correction  is  one-half  that  found  by  averaging  end  areas,  and 
has  the  opposite  sign,  i.  e.,  the  method  on  the  average  gives  results  too 
small,  but  the  error  is  only  one-half  as  great  as  by  the  preceding 
method. 

4.  Equivalent  Mean  Heights.— In  this  method  the  center  height 
of  a  level  section  having  an  equal  area  is  found  for  each  end  section, 


Fig.  2. 

and  the  volume  of  the  reduced  solid  is  then  accurately  found  by  the 
prismoidal  formula. 

Applying  this  to  two-level  ground,  with  slope  stake  heights  *„  ^r, 
slope  ratio  s,  reduced  center  height  clt  and  distances  out  to  slope  stakes 

**  8    J  * 


From  the  reduced  area, 
or; 


PRI&MOIDAL  AND  OTHER  FORMULAS.          [g  5, 


The  true  volume  of  the  reduced  solid, 
Vol.eim.hl  = 


The  true  volume  of  the  original  solid, 


Subtracting  the  volume  of  the  reduced  solid  from  that  of  the  origi- 
nal one, 


Vol.p-Vol.einihi=  [/,  */ 


or, 


Cor. 


e.  m.  hi 


(4) 


The  correction  is  thus  always  positive,  *,  <?.,  the  method  always  gives 
too  small  a  result  for  two-level  ground,  except  when  ^  //  =  //  tr ; 
hence  it  will  usually  give  too  small  results  for  ordinary  ground. 

5.  Triangular  Prismoid  -with  Curved  Center  Line. — In  curving 
the  center  line  of  the  roadbed  to  the  radius  J?,  let  the  dimensions  of 


Fig-3- 


Eq.  4]  CURVED  CENTER  LINE.  5 

all  the  cross  sections  remain  unchanged.  This  will  give  the  solid 
easiest  to  approximate  in  the  field,  and  the  one  most  like  the  prismoid, 
since  all  lineal  dimensions  of  the  cross  section  can  be  interpolated. 

In  Fig.  3,  let  B  C  be  the  center  line  of  the  roadbed  ;  e,  e<)>,  e1 ',  the 
eccentricities  of  the  sections,  or  the  distances  of  their  centers  of  grav- 
ity from  the  center  of  the  roadbed  ;  A,  AQ,  A/,  the  areas  of  the  trian- 
gular sections,  with  bases  b,  b<j>,  bf,  and  altitudes  a,  a<j>,  a' ;  $  the  cen- 
tral angle  from  B  to  any  point  F;  I  the  length  B  Cof  the  solid  along 
the  center  line  of  the  roadbed,  —  Rtyf . 

At  any  point  D, 

Elementary  volume  =  A^  D  E  =  A^  (R  -\-  e^)  dq> 

/     — R 
A.  (R 4-  e£d$ 
o 

Since  all  cross  section  dimensions  can  be  interpolated, 


Substituting, 
Volume  = 


--  +  b  (a'-a) 


+  [ab 


=\l  [ab  4-  a(b'—  b}  %  +  b(a'—a)  k  +  ( 


—a 


+[ab  ±  +a(b'-b)  i  +  d(a'—a)  i  +  (a'-a)(b'-b]  i  ]  ^^ 
=  ^\ab  +  i  ab'  +  i  a'b  +  a'b' 
+(ab  +  \ab'  +  \  a'b  +  a'V} 


PRISMOIDAL  AND  OTHER  FORMULAS.  \_\  5, 


m  m- 

Volume 

(5) 

where  Am  is  the  area  of  the  mid  section,  and  <?,  e'  ',  are  positive  for  the 
convex  side  of  the  curve. 

The  first  product  is  the  volume  which  would  be  obtained  by  the 
prismoidal  formula  neglecting  curvature,  leaving  the  second  for  the 
correction  due  to  curvature. 


Volume  for  Railway  Earthwork 

IN  CUBIC  YARDS  BY  AVERAGING  END  AREAS. 


6.  Formula  for  Three-level  Ground.  —Dimensions  in  feet,  volume 
in  cubic  yards. 

Area  of  section  =  (a+^r)-1 4.  (a+c)  -r  —a~ 


d, 


_    •         Fig.  4. 

At  the  next  section,  TOO  feet  from  the  first,  priming  corresponding 
parts, 

Area' =(«+»<)  ^-"y 

Multiplying  the  average  area  by  the  length,  and  dividing  by  27  to 
reduce  to  cubic  yards, 

/\     100 
Volume  =(area  +  area') 


2  X  27 

Volume  =(a+c]  w%%  —  a&%$  +  (a+c'}  w'  f  £  —  ab  f  £.  (6) 
/.  e.,  the  effect  of  each  cross  section  on  the  volume  reduces  to  the 
constant  times  the  total  width  between  slope  stakes  times  the  sum 
formed  by  adding  the  center  height  to  the  height  of  the  grade  trian- 
gle ;  minus  the  constant  times  the  base  times  the  altitude  of  the  grade 
triangle. 

For  the  next  solid, 

Volume'  =  (area'+  area")  i$£. 

The  terms  comprising  area'  should  therefore  be  added  separately, 
for  convenience  in  finding  the  second  volume,  etc. 

7.  Five -level  Ground.  —  In    five- level    ground    the    intermediate 

heights,  t{  and  tt,  are  taken  over  (or  under)  the  edges  of  the  roadbed. 

By  joining  the  edges  of  the  roadbed  with  the  ground  at  the  center,  two 

triangles  are  formed  on  the  left  having  /,  for  the  common  base,  rf,  for 

7 


8 


VOLUME  BY  AVERAGING  END  AREAS. 


the  sum  of  the  altitudes,  and  ^  dl  for  double  area  ;  two  on  the  right  with 
double  area  tt  dr ;  and  one  at  the  center  with  double  area  be. 


Therefore,     Area  first  end 


=*L^4.if*f  _{_*£ 


Area  second  end  = 


*L  +  **'**'  +  ^ 

222 


Volume  =  ^,f  |+^rff-h^f  l+A^/f  f+V^/f  t+^^f  f  .  (7) 
i.  e.,  the  effect  of  each  cross  section  on  the  volume  reduces  to  the  con- 
stant times  the  distance  out  to  the  left  slope  stake  times  the  left  shoul- 
der height  ;  plus  the  constant  times  the  distance  out  to  the  right  slope 
stake  times  the  right  shoulder  height  ;  plus  the  constant  times  the 
width  of  roadbed  times  the  center  height. 

8.  Irregular  Ground.  —  In  Fig.  6,  let  d\,  f\,  g\  •  •  •  ,  and  dr,  /r,  gr 
.  .  .  =  the  distances  to  the  left,  and  right,  to  the  slope  stakes,  and  in- 
termediate breaks  in  the  ground  surface  ;  and  r^  u^t\  .  .  .  ,  and  rr,  ur, 
tr  .  •  .  ,  =  the  corresponding  heights. 
H 


Fig-.  6. 


The  area  of  the  section  may  be  found,  by  taking  the  two  triangles 
with  the  common  base  a-\-c,  and  altitudes  </,  and  drt  adding  the  trian- 

i 

gle  with  area  GH-  --1,  and  subtracting  the  triangle  with  area  DE-  *J, 

and  the  grade  triangle  —     This  gives, 

Area  first  end  (the  right  side  being  regular) 


Kq.  8]  IRREGULAR  GROUND.  9 

where  IG  and  FE  are  to  be  taken  as  negative  if  measured  below  the 
horizontal  line  FI.     Expressing  them  in  terms  of  measured  quantities, 

since  IG  =  (r,—  c]  &  and  FE  =  (L—c]  £, 
d\  g\ 

Area  first  end 

W-,)+  etc. 


2  2 

—  (a  +  A)  — J  +  (u\  —  ri)  ^-  +  (^ — A)  •"  -f-  etc.    + 

2  2  '22 

where  each  break  or  intermediate  height  //,  Z>,  etc.,  adds  a  term  of  the 

form  (u{ — f\\—  •    If  //"is  omitted,  the  corresponding  term  («, — r,)^1  dis- 
2  2 

appears  and  the  other  term  becomes  (c—  rtf-1  ,  as  may  be  found  by 
leaving  off  the  triangle  GH&\  if  //"and  D  are  omitted  both  terms 

disappear,  and  (0+^)  J  becomes  (a+c)  -i,  as  may  be  seen  from  the  ex- 
pression for  the  right  side. 

If  the  next  cross  section,  TOO  feet  distant,  is  similar  to  the  first,  and 
the  same  letters  with  primes  are  used  for  corresponding  parts,  we  shall 
have, 

Area  second  end, 


=  (a  +  *,')    L+K'-^OH-  (*'-/I/)+  etc. 

222  22 

The  volume  will  be  found  by  multiplying  the  average  area  by  the 
length,  and  dividing  by  27  to  reduce  to  cubic  yards. 


Volume  in  cubic  yards  =  (area  4-  area7) 


100 


2  X  27 

Substituting  the  value  of  each  end  area, 
Volume 

etc.     +(a+c)dr 


(8) 
Or,  the  effect  of  each  cross  section  on  the  volume  reduces  : 

For  the  irregular  side  ;  to  a  constant  times  the  distance  to  the  slope 
stake  times  the  sum  formed  by  addiug  the  height  next  the  slope  stake 
to  that  of  the  grade  triangle  ;  pins  the  constant  times  the  distance  out 
to  each  break,  in  turn,  times  the  difference  formed  by  subtracting  the 
first  height  outside  from  the  first  height  inside  that  break. 

For  the  regular  side  ;  to  the  constant  times  the  distance  out  to  the 
slope  stake  times  the  sum  formed  by  adding  the  center  height  to  the 
height  of  the  grade  triangle- 


10  VOL UME  BY  A  VER AGING  END  AREAS.  \_\  9, 

From  the  total  sum  must  be  subtracted  the  constant  times  the  base 
times  the  altitude  of  the  grade  triangle. 
For  the  next  solid, 

Volume  =  (area'4-  area")  _!2?__  . 
2  X  27' 

the  terms  comprising  area'  should  therefore  be  added  separately,  for 
convenience  in  finding  the  second  volume,  etc. 

It  may  be  noted  that,  if  the  expressions  for  the  areas  of  the  origi- 
nal triangles  had  not  been  reduced,  the  constant  times  the  base  times 
the  altitude  of  each,  would  appear  as  its  effect  upon  the  volume. 
Hence  the  computer  may  divide  the  sections  into  any  triangles  he  may 
wish  ;  the  volume  will  result  by  multiplying  the  constant  into  the  base 
times  the  altitude  of  each  triangle,  and  adding  the  products. 

9.  Tables.— The  table  gives  the  value  of  (a-\-c)dk&  (or/rf|f),  for 
a  -f-  c  (height)  and  d  (width)  as  arguments  ;  or  it  gives  the  effect  which 
any  triangle  in  the  cross  section  has  upon  the  volume,  for  a  length  be- 
tween stations  of  100  feet,  by  using  the  altitude  and  base  of  the  trian- 
gle as  arguments.  After  looking  ont  the  partial  volume  for  each  tri- 
angle for  both  ends  and  adding,  if  the  length  be  not  TOO,  multiply  the 
sum  by  T^  of  the  length. 

To  look  out  the  partial  volume  for  a  triangle  : — Find  the  integral 
part  of  the  height  factor  in  the  line  of  height  headings  at  the  top  of  the 
page  ;  then  move  down  this  column  until  opposite  the  integral  part  of 
the  width  factor  in  the  width  column  and  note  the  number  ;  then  move 
to  the  left  on  this  line  (or  the  next  below  if  the  tenths  of  width  be 
greater  than  5)  and  take  out  the  "  Correction  for  tenths  of  height"  ; 
them  move  to  the  bottom  of  the  page  and  under  the  column  first  used 
(or  the  next  to  the  right  if  the  tenths  of  height  be  greater  than  5)  take 
out  the  "  Correction  for  tenths  of  width."  Take  the  sum  of  the  three 
mentally.  Look  out  the  quantity  for  each  of  the  other  terms  ;  the 
algebraic  sum  will  be  the  total  volume.  The  terms  coming  from 
"area  second  end,"  should  be  kept  separate  as  they  will  be  used  in 
finding  the  volume  of  the  next  solid. 

If  one  of  the  arguments  cannot  be  found  in  the  table  divide  it  by 
two,  look  out  the  number  and  multiply  by  two  ;  if  both  cannot,  divide 
each  by  two  and  multiply  by  four. 

In  order  to  impress  upon  the  mind  the  fact  that  the  correction  for 
tenths  of  width  is  placed  under  the  height  column,  it  should  be  remem- 
bered that  the  differences  down  the  column  are  constant,  so  that  cor- 
rections or  proportional  parts  of  these  differences  can  be  placed  under 
it,  but  the  widths  are  increased  as  we  move  down  the  column,  or  these 
corrections  are  for  tenths  of  width.  Similarly  for  the  lines  horizon- 
tally, or  the  corrections  for  tenths  of  height. 

After  becoming  familiar  with  the  table,  it  will  often  save  time  to  call 
height  "width,"  and  width  "height,"  and  to  take  the  correction  for 
tenths  from  the  side  or  bottom  of  the  page  as  most  convenient,  pro- 
vided always  that  it  is  opposite  (or  under)  the  heading  number  to 
which  it  belongs. 


Eq.  8]  EXAMPLE,  FIVE-LEVEL  GROUND.  n 

10.  Example,  Three-level  Ground. — 

Sta.  Center.  Left.  Right. 

20  .  .  10.8   . 


21 24.2,  ....  32-6 


19-6  '  24.1 

Hi!  32.6 

23-7' '  '  55-9 


21+20    .    .  46.7,  .  .,.  5Q.6 

37-0  82.9 

a  =4.7,  6  =  14.  f 

Substituting  in  formula  (6),  or  following  the  rule  $  6,  we  shall  have, 
in  connection  with  \  9  : 
Sta.  20, 

a  +  c  =  4.7  +  10.8  =  15.5,  w  =  19.6  +  24. i  =  43.7. 
From  the  Table  of  Triangular  Prismoids, 

Under  height  15,  right  of  width  43, 597 

Left  of  width  44,  correction  for  0.5,   ..........     20 

Under  height  15,  correction  for  0.7, 10 

627 

Arguments,  0  =  4.7  and  b  =•- 14  (grade  prism), — 61     566 

Sta.  21, 

a  +  ^'=4.7  +  24. 2  =  28.9,  ^'  =  23.7  +  55.9  =  79.6,   .  .    .    2131 

0  =  4.7,  £  =  14, — 6 1  2070 

Total  in  yards,  ...        2636 

From  above,  for  sta.  21, 2070 

Sta.  21  +  20, 

a  +  £'=4. 7  +  46. 7  =  51.4,  ze/'=37-o  +  82.9  =119.9,  5712 

a  =  4. 7,  £  =  14  (grade  prism), — 61  5651 

7721 
Total  in  yards 7721  X  .20=  1544 

Grand  total, 4180 

ii.  Example,  Five-level  Ground. — 

Sta.  Center.  Left.  Right. 

46  .  .  10.8,  .          .  IL4  .    .  £!_',  .  .  .  9^  .    .  A_6 

7  25.1  7  19.9 

47  -  -    6.4,  .          .  M  .    .  1±.  .  .  At  >  .'  K 

7  17-8  7  14-5 


*  The  numerator  is  the  height  above  or  below  grade  at  the  point  and 
the  denominator  is  its  distance  from  the  center. 

t  The  side  slope  s'  is  here  assumed  to  be  one  and  one-half  horizontal 
to  one  vertical,  so  that 


12  VOLUME  BY  A  VE RAGING  END  AREAS.        [g  12, 

Substituting  in  formula  (7),  using  the  Table  for  the  multiplications, 
Sta.  46,  arguments,  25.1  and  11.4,  265 
19.9     "      9.9,   183 
14.0     "    10.8,  140  588 

Similarly  for  Sta.  47,  272 

Total  in  yards,  860 

12.  Example,  Irregular  Ground. — 

Sta.         Center.  Left.  Right. 

20  .        .  10.8,  .  M,  ,  .  IL£  .    .  JH 

19.6  18          24.1 

21  24.2  IM  .    .  ^    .  20  4        32.6 

16         23.7'  42         55.9 

21+20  .  46.7,       .  4°iZ  .    .  2ao,  .      .  4*L£  .    i  _5°  .   .  50-6 

20  37.0  20  58  82.9 

b  =  14,  5  =  1.5  to  i,  a  =  4.7. 

Substituting  in  formula  (8),  using  the  Table  for  the  multiplications, 

Volume  from  sta.  20  to  sta.  2 1 . 

Sta.  20, 
a-\-t=  4.74-11.2=15.9,  o?r=24.i  ;  c—  rr=io.8 — 11.4= — 0.6,  £-r=i8  ; 

#+£=15.5,  ^,=19.6;  #=4.7,  £=14. 
Under  height  15,  right  of  width  24,  ..    .    .  333 

I/eft  of  width  24,  cor.  for  0.9, 20 

Under  height  16,  cor.  for  o. i, i 

354 

Arguments,  — 0.6  and  18, — 10 

19.6  and  15.5, 281 

14  and  4.7  (grade  prism),    ....      — 61 

•     564 
Sta.  21,  Arguments, 

20.4+4.7=25.1,  and  55.9,  ....  1299 
24.2 — 32.6= — 8.4,  and  42,  .  .  .  .  — 327 
18.6+4.7=23.3,  and  23.7,  ....  512 

24.2 — 11.1=13.1,  and  16, 194 

'         14  and  4.7  (grade  prism),     ....      — 61 

1617 

Total  in  yards, 2181 

Volume  from  21  to  21+20. 

From  above  for  sta.  21, 1617 

Sta.  21+20,     . 

50+4.7=54.7,  and  82.9, 4200 

48.1— 50.6=— 2.5,  and  58,  .    .    .    .    —134 

46.7— 50=— 3.3,  and  20, —62 

40.7+4.7=45.4,  and  37, 1556 

46.7 — 20=26.7,  and  20, 494 

14  and  4,7  (grade  prism),  ....  -  — 61 

5993 

7610 

76lO  X. 20=  1522 

3703 


Eq.  8]  SIDE  HILL    WORK.  13 

Any  other  method  of  dividing  the  sections  into  triangles  can  be  used 
if  preferred,  although  it  will  result  in  increased  labor. 

13.  Borrow  Pits. — In  borrow  pits,  ditches  or  other  work,  where  the 
cross  sections  are  made  up  of  triangles  or  trapezoids,  take  the  altitude 
and  base  of  each  triangle,  or  the  sum  of  the  parallel  sides,  and  the  per- 
pendicular distance  between  them  for  each  trapezoid,  as  arguments,  as 
with  the  two  sections  below. 


/^i">    -      '••"•  „      iX" 

^^^~,                                                                                                    M5£/^ 

•""vi      ;'0-<        a-  a          !           so            I/in 

s"-'- 

Sta.  I.»50. 

Sta.  i,  27  and  14  2,  . 

355 

50     "     28.0,  . 

1296 

50     "     26.4, 

1223 

20       "       12.6,    . 

233 

-   3107 

Similarly  for  sta.  1+50,  2672 

5779 

5779  X  -5<=>  =  2890  yards. 

14.  Sidehill  Work. — When  a  "grade  point"  occurs  in  a  cross  sec- 
tion, one  side  being  in  cut,  the  other  in  fill,  to  compute  the  cutting,  put 
in  a  break  at  the  grade  point  and  call  the  cuts  on  the  fill  side  zero,  and 
the  distance  to  the  slope  stake  b  -=-  2.  Then  proceed  as  for  irregular 
sections. 

In  computing  fills  the  process  would  be  reversed. 

EXAMPLE. 

Sta.         Center.  Left.  Right. 

46  .    .  -2.4,    .    .    .™.          ^ -  *4 

4.3  12.3  16.6 

A*J  J^  T      *7  _*2." 


Roadbed  18  feet  in  cut,   14  feet  in  fill,  slopes  \\  to  i.     Altitude  of 
ade  triangle  in  cut  =  9  -r-  —  —  6, 
For  cutting,  the  notes  would  be 


grade  triangle  in  cut  =  9  -v-  ii  —  6,  in  fill  =  7  -s-ll  =  4.7. 


46  . 

.     .  O.O,     . 

0.0 

2.2 

0.0 

An 

I  7 

4-3 

12.3 

3-8 

o.o 

9.0 

0.0 

'  14.7'  ' 

2.6 

9.0 

Giving, 

Sta.  46,  0.0+6.0=  6,  and  12.3,  .  .  69 
0.0—2.2==: — 2.2,  and  4.3,  .  — 9 
0.0+6.0=  6,  and  9,  ....  50 
Grade  triangle,  6  and  18,  — 100 

10 

Sta.  47,  similarly,    .    .        60 

Total  in  cu.  yards,  ...          7° 


VOLUME  BY  AVERAGING  END  AREAS. 


16 


Or  it  may  be  more  convenient  to  leave  out  the  grade  prism,  thus  : 
Sta.  46,  height=2.2,  width=  9    — 4.3=  4-7,  giving  -.59 
"     47,       "      =i-7,       "     =14-7+2.6=17.3,       "        .    .    28 
"      =3-8,       "       9.0.       "        .    .    32 


Total  in  yards,  . 
For  filling,  the  notes  would  be 
Sta.       Center.  Left. 

o.o  o.o 

4-3  7-o'  ' 


69 


Right. 


46  .    .  —2.4, 


47  - 

Giving, 
Sta.  46, 


o.o  o.o 

7.0'  2.6 

0.0+4.7=4.7,  and      7,  .    .      31 

2.4—0.0=2.4,     "     4.3,  .    .        9 

2.4+4.7=7.1,     "   16.6,  .    .     109 

Grade  triangle,  4.7      (<   14,  .    .  — 61 

Sta.  47,  similarly, 


-6.4 
16.6 

-  2-9 
11.4 


88 
n 


Total  fill  in  cu.  yards,  ...  99 

15.  End   Sections.  —  A   cross  section   should   be  taken   wherever 
either  edge  of  the  roadbed  in  cutting  reaches  grade.     This  will  require 
two  sections  in  passing  from 

a  cut  to  a  fill,  or  vice  versa, 
while  a  third  is  sometimes 
added  where  the  center  line 
reaches  grade.  The  solid,  in 
cut,  between  the  two  sections, 
is  a  pyramid  with  volume 
equal  to  area  of  base  into  one- 
third  the  altitude,  or  it  may 
be  taken  under  the  rule,  base 
into  one-half  the  altitude,  and 
afterwards  corrected  by  the 
prismoidal  formula. 

The  solid  in  fill,  between 
the  two  sections,  can  usually 
be  taken  as  having  the  same 
altitude,  the  volume  not  be- 
ing required  as  accurately  as 
for  cut  ;  the  exact  values  can 
be  interpolated  when  necessary. 

16.  Correction  for  Curvature.  —  For  a  solid  corresponding  to  the 
triangular  prismoid,  as  described  in  §  5,  we  have  from  (5), 


8. 


Eq.  13]  CORRECTION  FOR  CUR  VA  TURE.  1 5 

Correction  =  ^  \_(A  -f  2  AJ  e  +  (2  ^m  -f  ^4/)  */]  (9) 

'']  (approx.)  (10) 


The  latter  equation  is  the  one  in  common  use,  giving,  with  less  labor, 
sufficient  accuracy  for  earthwork  solids. 

Three-level  Ground. — To  find  e,  the  eccentricity  of  the  center  of 
gravity,  we  have, 

d,  dT 


Fig-  9 
v^' 

For  the  right  side, 


2 

Center  of  gravity  from  center  line  =  -i 

3 

(Product),  =  (a  +  c)  — r 
6 

Similarly,  for  the  left  side, 

(Product)!  =  — 


Hence  f (product),  +  (product), (a-\-c)        6 

total  area  ~  (a-f-^r)  dr  -\-  d\ 

Or,  ,-ftz£&  (ii) 

.       3 

Correction  =  J—\_A  (dr  —  dl)  -f  ^/7  (rf/  —  rf/)]  (12) 

6  /c 

where  ^4  and  ^x  must  include  the  grade  triangle. 

Or,  Correctiony  =  -^LQ  K  ~  ^i)  +  Qf  W  ~  ^/)]          ( 13) 

where  Q  and  Q'  are  the  tabular  quantities  or  volumes  corresponding  to 
A  and  A'  (A  =  f ^  Q)  and  Correction  y  is  in  cubic  yards. 

Five-level  and  irregular  ground  can  usually  be  taken  as  three-level, 
with  the  same  distances  to  slope  stakes,  for  the  purposes  of  this  cor- 
rection, unless  the  intermediate  heights  show  a  marked  tendency  to 
lie  above,  or  below,  the  lines  joining  the  side  heights  with  the  center, 
when  the  section  can  be  plotted  and  an  equalizing  line  drawn,  or  in 
extreme  cases  each  triangular  prismoid  can  be  taken  separately  by  (10). 

Sidehill  Ground,  neglecting  all  breaks  between  slope  stakes.  The 
center  of  gravity  will  be  on  the  line  joining  the  center  of  the  base  with 
the  vertex,  at  one-third  its  length  from  the  base. 


16  VOLUME  BY  AVERAGING  END  AREAS.         [§  17, 

The  distance  from  the  lower  end  of  this  line  to  the  center,  Fig.  10, 


2 

Distance  from  the  upper  end  =  dr. 

Hence  <?  =  f/_ M-f-i^r=  £ /_ f/j-|-^r\  (14) 

where  dl  will  be  positive  if  the  grade  point  is  on  the  same  side  of  the 
center  as  the  slope  stake. 
Substituting  in  (10), 

Correction  =  1-  £  A  (  b-  _  rf,  -f-  dt)  +  A'(b--  df  +  d/)~]      (15) 


Correction,,  =  _L  f  0  (  *  —  <*,  +  </r)  +  0'  (  *  —  */  +  <*/)  1 


2  _^ 

(16) 


17.  Example,  Correction  for  Curvature.  —  In  the  example  of  \  12, 
let  the  center  line  be  on  a  7°  curve  to  the  right,  ^  =  5730-1-7  =  819, 
and  let  the  intermediate  breaks  be  omitted  in  finding  the  correction 
for  curvature. 

Volume  from  sta.  2  j  to  sta.  21. 

Q,  including  the  grade  triangle,  =  625  ;  Qf  —  1678  ;  d\  =  19.6  ;  dr  = 
241  ;  flf/^23.7;  d/  =  55.9. 

Substituting  in  (13), 

Correction  —  —       _  [625(24.  1  —  19  6)4-1678(55.9—  23.7)]  =  23 

3X819 

Volume  from  sta.  21  to  sta.  21  +20, 

Correction  ==—J-_[[678(55.9—23.7)+6o54(82.9—  37.0)]  =  135 
3  X  819 

.2  X  135  27 

Total  in  cu.  yards  =      —  50 

The  correction  is  negative,  the  center  of  gravity  and  the  center  of 
the  curve  lying  on  the  same  side  of  the  center  line. 

In  the  example  of  \  1.4,  let  the  center  line  be  on  a  7°  curve  to  the  right. 
Volume  from  sta.  46  to  sta.  47. 
Substituting  in  (16), 

Correction  =  --  L_  [10(9—  12.3  —  4.3)  +  60(9  —  14.7  +  2.6)]  =  o 

3  X  °T9 

This  correction  would  be  positive,  the  center  of  gravity  and  the  cen- 
ter of  the  curve  being  on  opposite  sides  of  the  center  line. 


Prismoidal  Correction  in  Cubic  Yards. 


1 8.  Three-level  Ground. — The  prismoidal  correction  to  be  applied 
to  the  volume  of  a  triangular  prismoid  found  by  averaging  end  areas 
in  order  to  obtain  the  true  volume,  has  been  found  to  be  by  (2), 

Correction  =  (c  —  c'}(d'  —  d)~ 

where  c  and  d  are  the  altitude  and  base  of  the  triangle  at  the  first  end, 
and  c'  and  d'  those  at  the  second  end. 

For  three-level  ground,  per  station  in  cubic  yards,  this  would  re- 
duce to 

Pris.  correction  =  (c  —  c/)(zv/  —  w] — (17) 

3X54 

where,  as  in  Fig.  4  and  eq.  (6),  c  and  c'  are  the  center  heights  and  w 
and  w'  are  the  total  widths. 

This  correction  is  to  be  added  to  the  approximate  volume  found  by 
end  areas ;  it  disappears  when  the  center  heights,  or  the  widths,  are 
equal,  and  it  can  often  be  neglected  without  computation  when  in- 
spection shows  one  of  the  factors  very  small. 

19.  Five-level  Ground. — The  intermediate  heights  having  been  tak- 
en over  (or  under)  the  edges  of  the  roadbed.     In  $  7,  see  Fig.  5,  each 
section  was  divided  into  three  triangles,  one  having  the  distance  to  the 
left  slope  stake,  dlt  for  base,  and  the  height  over  the  left  edge  of  the 
roadbed,  tlt  for  altitude,  the  second  the   corresponding  dimensions, 
dr,  tr,  on  the  right,  and  the  third  the  width  of  roadbed  and  the  center 
height.     Applying  (2),  and  remembering  that  since  the  width  of  road- 
bed is  the  same  at  both  ends,  the  correction  for  the  third  triangle  will 
be  zero  : 

Pris.  correction  =|  (£-//)  (d/-di)+(tr-t')  (d'-dr)  IT^TT       (l8) 


20.  Irregular  Ground,  Borrow  Pits,  etc.  —  Since  the  areas  of  the 
sections  will  be  made  up  of  triangles,  the  correction  will  be  made  up  of 
as  many  terms  as  there  are  triangular  prismoids  in  the  volume,  each 
term  being  of  the  form 

(c  —  c/)(w/  —  w}     5° 


3X54 

This  requires  that  there  should  be  the  same  number  of  breaks  at  each 
end  of  the  solid,  as  already  indicated  in  $  i,  in  deriving  the  prismoidal 
formula  and  defining  the  prismoid.  If  there  are  not,  the  distance 
from  the  center  at  which  a  ridge  or  hollow  runs  out  at  the  more  regu- 
lar section  should  be  noted  in  the  field,  and  the  height  either  noted  or 
computed. 
17 


1 8     PRISMOIDAL  CORRECTION  IN  CUBIC  YARDS.    [§22, 

A.  M.  Wellington*  recommends  that  for  the  purposes  of  the  pris- 
moidal  correction  irregular  ground  be  assumed  to  be  three-level  ground 
having  the  center  height  and  total  width  of  the  irregular  section. 

This  makes  the  correction  the  same  as  for  regular  ground,  and  while 
for  a  single  volume  it  may  give  a  result  considerably  in  error,  the  sign 
is  as  likely  to  be  plus  as  minus,  so  that  there  is  less  tendency  to  accu- 
mulation of  error  than  in  the  various  approximate  methods  described 
in  ||  2-4 

21.  Table  of  Frismoidal  Corrections. — The  table  gives  the  value  of 

(c  —  c'}(w'  —  w]    5°    ,  for  c-cf  (height  —  height7)  and  w'  —  w  (width' 

3X54 

—width)  as  arguments  ;  which  is  the  correction  for  a  length  of  TOO  feet. 
If  the  length  be  not  100,  multiply  the  result  by  T^  of  the  length. 

To  look  out  the  number  : — Find  the  integral  part  of  the  height  fac- 
tor in  the  line  of  height  headings  at  the  top  of  the  page ;  then  move 
down  this  column  until  opposite  the  integral  part  of  the  width  factor 
in  the  width  column,  and  note  the  number ;  then  move  to  the  left  on 
this  line  (or  the  next  below  if  the  tenths  of  the  width  factor  be  greater 
than  5)  and  take  out  the  "  Correction  for  tenths  of  height — height'  "  ; 
then  move  to  the  bottom  of  the  page  and  under  the  column  first  used 
(or  the  next  to  the  right  if  the  tenths  of  the  height  factor  be  greater 
than  5)  and  take  out  the  "Correction  for  tenths  of  width' — width." 
Take  the  sum  of  the  three  mentally. 

If  one  of  the  arguments  cannot  be  found  in  the  table,  divide  it  by 
two,  look  out  the  number  and  multiply  by  two  ;  if  both  cannot,  divide 
each  by  two  and  multiply  by  four. 

The  remarks  of  |  9  apply  to  this  table  the  same  as  to  the  table  of 
triangular  prismoids. 

22.  Example,  Three-level  Ground. — See  \  10. 


Sta. 

Center. 

Left. 

Right. 

20  .    .    . 

.  .10.8  

8.4 

11.4 

19.6 

24.1 

21   .     .     . 

.  .24.2,  .    .    .    . 

ii.  i 

32.6 

23-7 

55-9 

2I+2O  . 

.  .46.7,  .    .    .    . 

20.  o 

50.6 

37-0 

82.9 

Correction  for  volume  from  sta.  20  to  sta.  21. 

c  —  c'=  10.8  —  24.2=  -  13.4,  w'  —  w  =  79.6  —  43.7  =  +  35.9. 
From  the  Table  of  Prismoidal  Corrections. 

Under  13,  right  of  argument  35,  =  140 

Left  36,  corr.  for  0.4  =     4 

Under  13,  corr.  for  0.9=     4  — 148 

*  Computation  from  Diagrams  of  Railway  Earthwork. 


Eq.  1 8]  EXAMPLE,  IRREGULAR  GROUND.  19 

Correction  for  volume  from  sta.  21  to  sta.  21  -f-  20. 
£  —  £'  =  24  2— 46.7  =  — 22.5,  w'—  ze>=  119,9,  —79-6  =  +  40.3; 
22.5  and  40.3  as  arguments  =  272  -\-  6  -f  2  =  280 

—  280  X  .20=  —  56 

Total  correction  to  the  volume  found  by  "  End  areas  "  =  — 204 

23.  Example,  Five-level  Ground. — 

Sta.  Center.  Left.  Right. 

46  .  10.8,        .  IM  .    .  "^  9^9      .  M 

8  20.1  8         16.6 

47  .  6.4,        .  M  .       7^  .  §^ .   .  1£ 

8  15.2'  8          13.0 

fy=  16,  side  slope  ratio 5=  i  to  i.     See  Fig.  5  and  (18). 

/,—  //  =  n.4  —6.8  =  +  4.6,  <//  —  ^,  =  15.2  —  20.1  =  —  4. 9. 

/r  —  //  =  9.9  —  5.7  =  +  4.2,  d/  —  dr  =  13.0  —  16.6  =  —  3.6. 
4.9  and  4.6  as  arguments  =  54-1  +  1  =  —  7 
4.2and3-6  =4+1         =—5 

Correction  in  yards  =  —12 

By  the  approximate  method  of  \  20  :    10.8  —  6.4  =  +  4.4  ;  28.2  — 
36.7=  — 8.5. 

+  4.4  and  —  8.5  =  10  -f  i  +  i  =  —  12,  =  correction  in  yards. 

24.  Example,  Irregular  Ground. — See  \  12. 


Sta. 

Center. 

Left. 

Right. 

20  .     . 

.   10.8,    . 

M, 

19.6'  " 

II.  2                 11.4 

'    18    '       '24.1 

18.6 

ii.  i 

20.4          32.6 

21 

.  24.2,  . 

'    16    ' 

w" 

42           '  55-9 

Aft  1 

40.7 

20.  o 

48.1            50.0           50.6 

21  -)-2O 

•  4v-7i  • 

20 

37.0 

20                      58                  82.9 

£  =  14,  .9  =  1.5  to  i,  a  =  4.7. 

The  break  16  feet  out  at  left  of  sta.  21  runs  out  10  feet  left,  at  sta.  20. 
The  break  20  feet  out  at  right  of  sta.  21+20  runs  out  at  the  center 

at  sta.  21. 
By  interpolation,  the  height  at  10  feet  to  the  left  at  sta.  20=  10.8  • 

J^_  [10.8  —  8.41=9.6. 
19.6 

Correction  for  volume  from  sta.  20  to  sta.  21. 

Sta.  20  would  be  written 

9-6  8.4 


..   .   . 

I0'"  '   *  lo    '       '  TP'  '  l8  24-x 

Changing  the  factors  of  §  12  which  were  used  in  finding  volume    or 
the  areas  of  the  triangles,  to  correspond  with  the  interpolated  break, 


20     PRISMOIDAL  CORRECTION  IN  CUBIC  YARDS.   [§25, 

Arguments,  19.6—23.7=—  4.1,23.3—14.3=+  9.0,  corr.=—   n 
24-1— 55-9=— 3i-8,  25.1— 15.9=+  9.2,     "        —  90, 
10    — 16    = —  6.0,  13.1 —  2.4=+io.7,     "        —  20 
"  18    —42    =—24.0,  —8.4-fo.6=—  7.8,     "        +  58  —63 

Correction  for  volume  from  sta.  21  to  sta.  21+20. 
Sta.  21  would  be  written 

24.2           l8-6              "-1             24.2  20.4             32.6^ 

16            '  23.7'  '            o  42              55.9' 
giving  as  above, 

Arguments,  23.7— 37.0=— 13.3,  45.4— 23.3=4-22.1,  corr.=—  90 

55-9— 82.9=— 27.0,  54-7— 25- 1=4-29-6,  "  —247 

16    —20    =—  4.0,  26.7— I3.i=+i3.6,  "  —.17 

42    —58    =—16.0,— 2.5+8.4=+  5.9,  "  —  29 

o    —20    =—2o.o,— 3.3— 3.8=—  7.1,  "  4  44 

—339 
339  X.  20=  —68 

Total  correction  in  yards=  — 131 

By  the  approximate  method  of  \  20  the  result  would  be  that  found 
for  the  first  example,  or  —  204  cu.  yds. 

25.  Example,  Borrow  Pits.— See  example,  \  13. 

Arguments  which  were  used  in  finding  volume ;  for  sta.  i,  27  and 
14.2,  50  and  28.0,  50  and  26.4,  20  and  12.6  ;  for  sta.  i  4  5°>  34  and  IO-4> 
50  and  21.4,  50  and  24.6,  17  and  13.6. 

The  corrections  will  all  disappear  except  for  the  first  and  last  pro- 
duct at  each  station. 

Arguments,  27  —  34  =  —  7,  10.4 — 14.2  =  —  3.8,  corr.  =  +  8 
"  20— 17  =  +  3,  13.6  — 12.6  = +  1.0,     "     =  +  i 


Correction  in  yards  =  9  +  .50  =  5. 


Computation— End  Areas  with  Prismoidal 
Correction. 


26.  Three-level  Ground. — 

St 
o 


..  Center. 

7  I 

Left. 
6-4 

Right. 
8.0 

.     9  8 

14.4 
7-6 

16.0 
12.4 

.  12.4  . 

'  15-6 
12.4 

20.4 
10.3 

8  7 

20.4 
9-8 

18.3 
8.1 

.     .   IO.2  

'  17.8  ' 

u.  6 
19.6 

16.1 

9.8 

'  17.8 

4 
Roadbed  16  ft.,  side  slopes  i  to  I. 

COMPUTATION. 


Sta. 

Width. 

Height, 
orh.+a. 

Yds.  by  Knd  areas 

w  —  w'. 

ht'  —  ht. 

Yards, 
pris'dl. 

o 

i 

2 

3 
4 

30-4 
36.0 

38.7 
33-9 
37-4 

I5-I 
17.8 
20.4 

16.7 
18.2 

426 
£  594 

M       73I 
525 
631 

782 
1087 

1018 
918 

-5.6 
—2.7 

+4-8 
—3-5 

+2.7 

+2.6 

—3-7 

+  1-5 

—5 

—  2   ' 

—6 

—  2 

Approx.  volume  =  3805 
Prismo'l  cor.   =  —      15 

True  volume  =       3790 


27.  Irregular  Ground.— 


5-2 


7-4         3-4 


V  • 

13.2          8 

11.4 

IO  . 

.  4.9  • 



9-6         4.1 
17.6         9    ' 

3-6 
ii.  6 

7-2           5.0 

8.4         5-2 

8.9 

10,14  1  * 

.  9.0  . 

8                   12 

16.4         7 

16.9 

16     |  12 

12 

6.8 

11.4 

12.8 

7-1 

16 

20.8 

15.1 

T1 

8  6 

7-7 

7-4 

5-2 

M  * 

12 

15-4 

13.2 

*The  breaks  at  the  left  of  n  run  out  10  ft.  and  14  ft.  respectively  to 
the  left  of  10,  etc. 


22     END  AREAS  WITH  PRISMDL.  CORRECTION. 


27, 


+47  •    -  2.o  . 

4 

5-4 

0.0 

10 

13-4 

8.0 

0.0 

4-2 

2.0 

1 

4.o 

12.2 

IO.O 

8| 

+2O  .  —  6.O    .     .     . 

O.O 

4.4 

8.0    ' 

13-6 

Roadbed  16  ft,  slopes  i  to  I  in  cut. 
Roadbed  14  ft.,  slopes  i^  to  i  in  fill. 


Computation  on  the  assumption  that,  for  the  purpose  of 
moidal  correction  only,  the  ground  is  regular. 


the  pris- 


Sta. 

Width. 

Height, 
or  h.+a. 

Yds.  by  Bnd  areas 

w  —  w'. 

ht'—  ht. 

Yards, 
pris'dl. 

9 

13.2 

148 

182 

11.4 

15-4 

163 

8 

3-4 

25 

10 

17.6 

12.9 

211 

1  1.6 

12.  1 

130 

9 

i-3 

II 

484 

-4.6 

—1.9 

+  2 

ii 

16.4 

13.0 

I98 

12. 

—  1.2 

—13 

8 

+4 

30 

16.9 

13.2 

ON     207 

7 

+0.1 

M                I 

537 

—4.1 

+4-1 

—5 

12 

20.8 

19.4 

1    374 

16.0 

—6.0 

-89 

15-1 

14.8 

206 

676 

—2.6 

—  2.2 

+  i 

13 

15.4 

15-7 

224 

12. 

1.2 

13 

13.2 

16.6 

203 

693 

+7.3 

+  1.8 

+4 

+47 

13.4 

12.0 

148 

10. 

—3-4 

—32 

8 

IO.O 

_    74 

184 

+7-2 

—6.6 

—  7 

Pris'l 
for  14 

r 

5-4 

o 

14 

4.0 

4.2 

1      16 

46 

+  12.0 

—  1.2 

—3 

+20 

O.O 

0.0 

oo 

3 

+  4.0 

—4.2 

—  i 

Approx.  volume  =  2623 
Prismo'l  cor.    —  —        9 


Corrected  volume  =       2614 


—  16 


—  9 


Eq.  18] 


RULES  FOR  CROSS  SECTIONING. 


Computation—  With  the  true  prismoidal  correction. 


Sta. 

Width. 

Height, 
or  h.+a. 

Yds.  by  End  areas 

w  —  w'. 

ht'—  ht. 

Yards, 
pris'dl. 

9 

13.2 

14.8 

182 

11.4 

15.4 

163 

8 

3-4 

25 

10 

17.6 

12.9 

211 

—4.4 

—1.9 

+2 

Pris'l* 

Ji7.6 

16.6 

for  ii 

1  J^" 

—1.9 

1  10 

—3-7 

n.6 

I2.I 

I30 

—  0.2 

—3-3 

9 

1-3 

II 

484 

—  1.0 

—  2.1 

+i 

ii 

16.4 

13.0 

I98 

+  1.2 

-3-6 

—  i 

12 

—  1.2 

—13 

+2.0 

+0.7 

8 

+4.0 

30 

+2.0 

+7-7 

+4 

16.9 

13.2 

2O7 

—5-3 

+1.1 

—  2 

7 

+0.1 

I 

537 

+2.0 

—  1.2 

—  I 

12 

20.8 

19.4 

374 

—4-4 

+6-4 

—8 

16 

—6.0 

oT~89 

Pris'l* 

/  J6 

—1.4 

—4.0 

—  O.2 

for  ii 

Ii6 

-4.6 

H 

I 

—8.0 

—8.6 

+21 

15.1 

14.8 

1    206 

Pris'l* 

/i5-i 

15.0 

+1.8 

+1.8 

+  1 

for  ii 

1  12 

—o-3 

676 

—5-0 

—0.4 

+  1 

13 

15-4 

15.7 

224 

+5-4 

—3-7 

—6 

12 

1.2 

13 

+4-o 

+7-2 

+9 

13.2 

16.6 

203 

693 

+1-9 

+1.8 

+47 

13-4 

12.0 

148 

+2.0 

—3-7 

—  i 

IO 

—3-4 

—32 

+  2.0 

—4.6 

—  i 

8 

IO.O 

74 

184 

+5-2 

—6.6 

—5 

14 

12.2 

8.0 

90 

+  1.2 

—4.0 

i 

4 

—42 

—  16 

+6.0 

—0.8 

i 

8 

8.0 

59 

45 

0.0 

—  2.O 

+20 
Pris'l* 

(    8 
1    8 

8.0 

0.0 

+4.2 

—4.0 

O.O 

+4.2 

—  i 

for  14 

I   8 

8.0 

3 

0.0                0.0 

Approx.  volume  +  2622                                         +  40 

Prism'l.  Cor.   =+12                                         —28 

True  volume        =  2634 

28.  Rules  for  Cross  Sectioning. — In  deriving  the  prismoidal  for- 
mula, the  ground  surface  of  each  prismoid  was  assumed  to  be  a  hyper- 
bolic paraboloid.  Hence  if  the  prismoidal  formula,  or  its  equivalent, 
is  to  be  used  : 


*  These  can  usually  be  avoided  in  practice  by  placing  the  same  num- 
ber of  breaks  in  adjacent  sections  as  already  indicated  in  $  20. 


24  RULES  FOR  CROSS  SECTIONING.  [§  28, 

Place  cross  sections  at  the  regular  stations  (usually  100  feet  apart), 
treating  the  ground  as  three-level  if  the  surface  is  very  regular ;  five- 
level  if  at  all  broken,  as  the  increased  labor  in  field  and  office  is  slight ; 
and  irregular  if  more  broken.  The  object  being  to  obtain  points 
enough  in  each  cross  section  so  that  straight  lines  joining  them  will 
lie  in  the  surface,  or  will  equalize  the  surface,  and  to  adopt  the  meth- 
od which  will  secure  this  at  most  of  the  sections  without  extra  breaks, 
even  if  it  gives  more  points  than  are  necessary  at  some  of  the  more 
regular  stations. 

An  occasional  extra  break  can  often  best  be  treated  by  itself,  as  add- 
ing or  subtracting  a  computed  amount  to  or  from  the  regular  quantity 
which  would  be  obtained  without  it. 

It  should  also  be  noted  that  the  formula  requires  the  same  number 
of  sides  or  points  in  adjacent  cross  sections  (dividing  the  solid  into  tri- 
angular prismoids),  requiring  that  when  a  ridge  or  hollow  runs  out  its 
vanishing  point  in  the  more  regular  section  be  taken  and  the  section 
be  made  up  with  the  break  for  the  solid  on  one  side  and  without  it  for 
the  solid  on  the  other  side  of  the  section.  The  height  of  this  vanish- 
ing point  can  be  computed,  but  it  is  usually  better  to  measure  it  in  the 
field. 

Intermediate  cross  sections  should  be  added  whenever  straight  lines 
joining  corresponding  points  of  the  regular  sections  will  not  lie  in  the 
surface  or  equalize  the  surface,  and  enough  should  be  added  to  fulfill 
this  condition.  When  the  center  line  is  curved,  the  lines  joining  cor- 
responding points  of  adjacent  cross  sections  should  be  estimated  at  the 
same  curvature  in  judging  of  their  coincidence  with  the  ground  sur- 
face. 

A  cross  section  should  also  be  placed  wherever  a  grade  point  occurs 
on  either  edge  of  the  roadbed  (usually  limited  to  the  edges  in  cutting) 
and  one  is  often  placed  where  a  grade  point  occurs  on  the  center  line. 
In  light  work  and  where  the  transverse  slope  is  small,  a  section  is 
sometimes  taken  where  the  center  line  comes  to  grade  and  the  other 
two  sections  omitted.  See  $  15. 

The  plus  stations  are  preferably  placed  at  50  feet,  at  multiples  of  10, 
or  at  whole  feet  (except  grade  points)  from  the  last  full  station,  for 
convenience  of  computing  grades  in  the  field  and  quantities  in  the 
office.  Breaks  on  a  cross  section  should  be  put  at  multiples  of  10,  or 
at  whole  feet,  from  the  center,  when  possible  without  a  sacrifice  of  ac- 
curacy. 

When  the  method  by  averaging  end  areas  is  used,  without  the  pris- 
moidal  correction,  additional  sections  should  be  taken  whenever  the 
change  in  transverse  slope  or  in  area  between  adjacent  sections  is 
great.  The  reason  for  this  can  be  seen  by  referring  to  (2)  ;  halving 
the  distance  between  sections  will  halve  both  factors  (c  —  cf}  and 
(w'  —  w],  or  quarter  the  product,  so  that  when  the  two  half  length 
solids  are  added  together  the  correction  or  error  will  be  reduced  one- 
half.  The  error  due  to  averaging  end  areas  can  thus  be  made  as  small 
as  desirable,  but  the  method  if  extended  far  would  soon  involve  more 
extra  labor  than  to  apply  the  prismoidal  correction. 

The  same  remarks  will  apply  to  the  method  by  middle  areas. 


TABLE 


—  OF  — 


Triangular  Prismoids  for  Railway  Earthwork 


IN    CUBIC   YARDS, 
BY  AVERAGING  END  AREAS. 

(See  Prismoidal  Correction.) 


[Width  1-50:  Ht.  i-io.J 


VOL UME  BY  A  VERA GING 


.7 

2 

3 

4 

5 

6 

7 

8 

9 

WIDTH. 

1 

2 

345 

678 

9   10 

i 

i 

i 

i 

I 

i 

2 

3 

4    5 

6 

6    7 

8|    9 

I 

i 

i 

i 

i 

i 

2 

2 

2 

4 

6 

7    9 

ii 

13   15 

17 

19 

i 

i 

i 

2 

2 

2 

3 

3 

3 

6 

8   ii 

14 

17 

19    22 

25 

28 

i 

i 

2 

2 

3 

3 

3 

4 

4 

7 

n 

15 

19 

22 

26    30 

33 

37 

i 

2 

2 

3 

3 

4 

4 

5 

5 

9 

H 

19 

23 

28   32   37 

42 

46 

2 

2 

3 

3 

4 

4 

5 

6 

6 

n 

17 

22 

28 

33 

39 

44 

'50 

56 

2 

3 

3 

4 

5 

5 

6 

7 

6 

13 

19 

26 

32 

39 

45 

52 

58 

65 

2 

3 

4 

4 

5 

6 

7 

8 

7 

15 

22 

30 

37 

44 

52 

59 

67 

74 

2 

3 

3 

4 

5 

6 

7 

8 

9 

8 

17 

25 

33 

42 

50 

58 

67 

75 

83 

2 

3 

4 

5 

6 

6 

7 

8 

IO 

9 

28 

37 

46 

56 

65 

74 

83 

93 

2 

3 

4 

5 

6 

7 

8 

9 

ii 

IO 

20 

31 

41 

5I 

61 

7i 

8  1 

92 

IO2 

2 

3 

4 

6 

7 

8 

9 

IO 

12 

ii 

22 

33 

44 

56 

67 

78 

89 

IOO 

III 

2 

4 

5 

6 

7 

8 

10 

n 

13 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

3 

4 

5 

6 

8 

9 

10 

12 

14 

13 

26 

39 

52 

65 

78 

91 

104 

117 

130 

3 

4 

6 

7 

8 

10 

II 

12 

15 

H 

28 

42 

56 

69 

83 

97 

in 

125 

139 

I 

3 

4 

6 

7 

9 

10 

12 

13 

16 

15 

30 

44 

59 

74 

89 

104 

119 

133 

148 

2 

3 

5 

6 

8 

9 

ii 

13 

17 

16 

31 

47 

63 

79 

94 

no 

126 

142 

157 

2 

3 

5 

7 

8 

10 

12 

13 

15 

18 

17 

33 

50 

67 

83 

IOO 

117 

133 

150 

I67 

2 

4 

5 

7 

9 

ii 

12 

H 

16 

19 

18 

35 

53 

70 

88 

1  06 

123 

141 

158 

I76 

2 

4 

6 

7 

9 

n 

13 

15 

17 

20 

19 

37 

56 

74 

93 

in 

130 

148 

167 

2 

4 

6 

8 

10 

12 

M 

16 

18 

21 

Ig 

39 

58 

78 

97 

117 

136 

156 

175 

194 

2 

4 

6 

8 

10 

12 

14 

16 

18 

22 

20 

61 

81 

102 

122 

'43 

163 

183 

2O4 

2 

4 

6 

9 

ii 

13 

15 

17 

19 

23 

21 

43 

64 

85 

1  06 

128 

149 

170 

192 

213 

2 

4 

7 

9 

n 

13 

16 

18 

20 

24 

22 

44 

67 

89 

III 

133 

156 

178 

200 

222 

2 

5 

7 

9 

12 

16 

19 

21 

25 

23 

46 

69 

93 

116 

139 

162 

185 

208 

231 

2 

5 

7 

10 

12 

I4 

17 

!9 

22 

26 

24 

48 

72 

96 

120 

144 

169 

193 

217 

241 

3 

5 

8 

10 

12 

15 

18 

20 

23 

27 

25 

50 

75 

IOO 

125 

150 

175 

200 

225 

250 

3 

5 

8 

10 

13 

16 

18 

21 

23 

28 

26 

52 

78 

104 

130 

156 

181 

207 

233 

259 

3 

5 

8 

II 

13 

16 

'9 

21 

24 

29 

27 

54 

81 

107 

134 

161 

1  88 

215 

242 

269 

3 

6 

8 

II 

17 

'9 

22 

25 

30 

28 

56 

83 

in 

167 

194 

222 

250 

278 

3 

6 

9 

11 

14 

17 

20 

23 

26 

31 

29 

57 

86 

115 

144 

172 

201 

230 

258 

287 

3 

6 

9 

12 

15 

18 

21 

24 

27 

32 

30 

59 

89 

119 

I48 

178 

207 

237 

267 

296 

3 

6 

9 

12 

15 

18 

21 

24 

28 

33 

31 

61 

92 

122 

153 

183 

214 

244 

275 

306 

3 

6 

9 

13 

16 

19 

22 

25 

28 

34 

63 

94 

126 

157 

189 

22t 

252 

283 

3J5 

3 

6 

IO 

13 

16 

19 

23 

26 

29 

35 

32 

65 

97 

130 

162 

194 

227 

259 

292 

324 

3 

7 

IO 

13 

17 

20 

23 

27 

30 

36 

33 

67 

IOO 

133 

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12 

1  8 

23 

29 

35 

4i 

47 

53 

63 

2392 

2450 

2508 

2567 

2625 

2683 

2742 

2800 

2858 

2917 

6 

12 

18 

24 

3° 

36 

4i 

47 

53 

64 

2430 

2489 

2548 

2607 

2667 

2726 

2785 

2844 

2904 

2963 

6 

12 

18 

24 

30 

36 

42 

48 

54 

65 

2468 

2528 

2588 

2648 

2708 

2769 

2829 

2889 

2949 

3009 

6 

12 

18 

24 

3i 

37 

43 

49 

55 

66 

2506 

2567 

2628 

2689 

2750 

2811 

2872 

2933 

2994 

3056 

6 

12 

19 

25 

31 

37 

43 

5o 

56 

67 

2544 

2606 

2668 

2730 

2792 

2854 

2916 

2978 

3040 

3102 

6 

13 

J9 

25 

31 

38 

44 

50 

57 

68 

2581 

2644 

2707 

2770 

2833 

2896 

2959 

3022 

3085 

3148 

6 

13 

T9 

26 

32 

3« 

45 

5i 

58 

69 

2619 

2683 

2747 

28ll 

2875 

2939 

3°o3 

3067 

3*3! 

3J94 

6 

13 

19 

26 

32 

39 

45 

52 

58 

70 

2657 

2722 

2787 

2852 

2917 

2981 

3046 

3111 

3176 

3241 

7 

!3 

20 

26 

33 

39 

46 

53 

59 

7i 

2695 

2761 

2827 

2893 

2958 

3024 

3090 

3156 

3221 

3287 

7 

J3 

20 

27 

33 

40 

47 

53 

60 

72 

2733 

2800 

2867 

2933 

3000 

3067 

3i33 

3200 

3267 

3333 

7 

14 

20 

27 

34 

4r 

47 

54 

61 

73 

2771 

2839 

2906 

2974 

3042 

3109 

3i77 

3244 

3312 

338o 

7 

14 

21 

27 

34 

4i 

48 

55 

62 

74 

2809 

2878 

2946 

3015 

3083 

3152 

3220 

3289 

3357 

3426 

7 

14 

21 

28 

35 

42 

49 

56 

63 

75 

2847 

2917 

2986 

3056 

3125 

3i94 

3264 

3333 

3403 

3472 

7 

14 

21 

28 

35 

42 

49 

56 

63 

76 

2885 

2956 

3026 

3096 

3167 

3237 

3307 

3378 

3448 

3519 

7 

14 

21 

29 

36 

43 

50 

57 

64 

77 

2923 

2994 

3066 

3137 

3208 

3280 

335i 

3422 

3494 

3565 

7 

14 

22 

29 

36 

43 

5i 

58 

65 

78 

2961 

3033 

3106 

3178 

3250 

3322 

3394 

3467 

3539 

3611 

7 

15 

22 

29 

37 

44 

5i 

59 

66 

79 

2999 

3072 

3145 

3219 

3292 

3365 

3438 

35ii 

3584 

3657 

7 

15 

22 

30 

37 

44 

52 

59 

67 

80 

3037 

3III 

3185 

3259 

3333 

3407 

348i 

3556 

3630 

3704 

8 

15 

23 

30 

37 

45 

53 

60 

68 

81 

3075 

3150 

3225 

3300 

3375 

3450 

3525 

3600 

3675 

3750 

8 

15 

23 

30 

38 

46 

53 

6r 

68 

82 

3Ji3 

3^9 

3265 

3341 

3417 

3493 

3569 

3644 

3720 

3796 

8 

15 

23 

31 

38 

46 

54 

61 

69 

83 

3J5i 

3228 

3305 

3381 

3458 

3535 

3612 

3689 

3766 

3843 

8 

16 

23 

31 

39 

47 

54 

62 

70 

84 

3189 

3267 

3344 

3422 

35oo 

3578 

3656 

3733 

3811 

3889 

8 

16 

24 

31 

39 

47 

55 

63 

7i 

85 

3227 

33°6 

3384 

3463 

3542 

3620 

3699 

3778 

3856 

3935 

8 

16 

24 

32 

40 

48 

56 

64 

72 

86 

3265 

3344 

3424 

3504 

3583 

3663 

3743 

3822 

3902 

398i 

8 

16 

24 

32 

40 

48 

56 

64 

73 

87 

3303 

3383 

3464 

3544 

3625 

3706 

3786 

3867 

3947 

4028 

8 

16 

24 

33 

4i 

49 

57 

65 

73 

88 

334i 

3422 

3504 

3585 

3667 

3748 

3830 

3911 

3993 

4074 

8 

16 

25 

33 

4i 

49 

58 

66 

74 

89 

3379 

346i 

3544 

3626 

3708 

3791 

3873 

3956 

4038 

4120 

8 

17 

25 

33 

42 

50 

58 

67 

75 

90 

3417 

35oo 

3583 

3667 

3750 

3833 

3917 

4000 

4083 

4167 

8 

17 

25 

34 

42 

5i 

59 

67 

76 

91 

3455 

3539 

3623 

3707 

3792 

3876 

3960 

4044 

4129 

4213 

9 

17 

26 

34 

43 

5i 

60 

68 

77 

92 

3493 

3578 

3663 

3748 

3833 

3919 

4004 

4089 

4174 

4259 

9 

17 

26 

34 

43 

52 

60 

69 

78 

93 

353  i 

3617 

3703 

3789 

3875 

3961 

4047 

4133 

4219 

4306 

9 

17 

26 

35 

44 

52 

61 

70 

78 

94 

3569 

3656 

3743 

3830 

3917 

4004 

4091 

4178 

4265 

4352 

9 

18 

26 

35 

44 

53 

62 

70 

79 

95 

3606 

3694 

3782 

3870 

3958 

4046 

4134 

4222 

43  10 

4398 

9 

18 

27 

36 

44 

53 

62 

7i 

80 

96 

3644 

3733 

3822 

39" 

4000 

4089 

4178 

4267 

4356 

4444 

9 

18 

27 

36 

45 

54 

63 

72 

81 

97 

3682 

3772 

3862 

3952 

4042 

4I31 

4221 

4311 

4401 

449  i 

9 
9 

CO  CO 

27 
28 

36 

37 

45 
46 

54 
55 

64 
64 

73 
73 

82 

83 

98 
99 

3720 
3758 

3811 
3850 

3902 
3942 

3993 
4033 

4083 
4125 

4174 
4217 

4265 
4308 

4356 
4400 

4446 
4492 

4537 
4583 

9 

r9 

28 

37 

46 

56 

65 

74 

83 

IOO 

3796 

3889 

398i 

4074 

4167 

4259 

4352 

4444 

4537 

4630 

.7 

.2 

-3 

-4 

-5 

.6 

•  7 

.8 

•9 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

O.I 
0.2 

8 

8 

8 

8 

8 

4 
9 

4 
9 

4 
9 

5 
9 

5 
9 

°-3 

ii 

12 

12 

12 

13 

13 

13 

13 

14 

14 

Correction  for  tenths 

0.4 

15 

16 

16 

16 

17 

17 

17 

18 

18 

19 

0.5 

19 

19 

20 

20 

21 

21 

22 

22 

23 

23 

of  width. 

0.6 

23 

23 

24 

24 

25 

26 

26 

27 

27 

28 

0.7 

27 

27 

28 

29 

29 

30 

30 

31 

32 

32 

0.8 

3° 

31 

32 

33 

33 

34 

35 

36 

36 

37 

0.9 

34 

35 

36 

37 

38 

38 

39 

40 

41 

42 

(II) 


TABLE 

—  OF  — 

Prismoidal  Corrections  for  Triangular 
Prismoids  for  Railway  Earthwork, 

IN   CUBIC   YARDS, 

To  be  applied  to  the  approximate  volume  found 
by  averaging  end  areas. 


[wx  — w  1^50  :  h  —  h'  i-io.] 


PRISMOIDAL 


.1 

2 

j 

# 

5 

<5 

7 

8 

9 

w'—  w. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10, 

I 

i 

i 

i 

2 

2 

2 

2 

3 

3 

i 

2 

i 

i 

2 

2 

3 

4 

4 

5 

6 

6 

i 

i 

i 

i 

3 

i 

2 

3 

4 

5 

6 

6 

7 

8 

9 

i 

i 

i 

i 

4 

i 

2 

4 

5 

6 

7 

9 

IO 

ii 

12 

I 

i 

i 

i 

i 

5 

2 

3 

5 

6 

8 

9 

ii 

12 

14 

15 

i 

i 

i 

i 

2 

6 

2 

4 

6 

7 

9 

ii 

13 

15 

17 

19 

i 

i 

2 

2 

2 

7 

2 

4 

6 

9 

ii 

13 

15 

17 

19 

22 

i 

i 

2 

2 

2 

8 

2 

5 

7 

10 

12 

15 

17 

2O 

22 

25 

i 

i 

i 

2 

2 

2 

2 

9 

3 

6 

8 

ii 

14 

17 

19 

22 

25 

28 

I 

i 

2 

2 

2 

2 

3 

10 

3 

6 

9 

12 

15 

19 

22 

25 

28 

31 

i 

2 

2 

2 

3 

3 

ii 

3 

7 

10 

14 

17 

20 

24 

27 

31 

34 

i 

2 

2 

3 

3 

3 

12 

4 

7 

ii 

15 

19 

22 

26 

30 

33 

37 

i 

2 

2 

2 

3 

3 

4 

13 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

i 

2 

2 

3 

3 

3 

4 

14 

4 

9 

J3 

17 

22 

26 

3° 

35 

39 

43 

i 

2 

2 

3 

3 

4- 

4 

15 

5 

9 

14 

19 

23 

28 

32 

37 

42 

46 

i 

2 

2 

3 

3 

4 

4 

16 

5 

10 

15 

20 

25 

30 

35 

40 

44 

49 

I 

2 

2 

3 

3 

4 

4 

5 

17 

5 

10 

16 

21 

26 

31 

37 

42 

47 

52 

I 

2 

2 

3 

3 

4 

4 

5 

18 

6 

ii 

17 

22 

28 

33 

39 

44 

50 

56 

I 

2 

2 

3 

4 

4 

5 

19 

6 

12 

18 

23 

29 

35 

41 

47 

53 

59 

I 

2 

2 

3 

4 

4 

5 

6 

20 

6 

12 

19 

25 

31 

37 

43 

49 

56 

62 

I 

2 

3 

3 

4 

5 

5 

6 

21 

6 

13 

19 

26 

32 

39 

45 

52 

58 

65 

I 

2 

3 

3 

4 

5 

5 

6 

22 

7 

14 

20 

27 

34 

4i 

48 

54 

61 

68 

I 

i 

2 

^ 

4 

4 

5 

6 

6 

23 

7 

u 

21 

28 

35 

43 

50 

57 

64 

7i 

I 

i 

2 

3 

4 

4 

5 

6 

7 

24 

15 

22 

30 

37 

44 

52 

59 

67 

74 

I 

2 

2 

3 

4 

5 

5 

6 

7 

25 

8 

15 

23 

31 

39 

46 

54 

62 

69 

77 

I 

T 

2 
2 

2 

3 

•^ 

4 

5 

5" 

6 
6 

6 

7 

26 

27 

8 
8 

16 

24 

32 

40 

48 

CQ 

56 
«;8 

64 
6? 

72 

7c 

80 
83 

I 

2 

3 

2 

4 

c 

6 

7 

8 

•W 

28 

9 

^7 

26 

35 

43 

J»^ 

52 

Du 
60 

w 
69 

/«. 

78 

86 

I 

2 

3 

A 

4 

§ 

6 

7 

8 

29 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

I 

2 

4 

5 

6 

6 

7 

8 

30 

9 

19 

28 

37 

46 

56 

65 

74 

83 

93 

I 

2 

4 

5 

6 

7 

8 

9 

3i 

10 

19 

29 

38 

48 

57 

67 

77 

86 

96 

I 

2 

4 

5 

6 

7 

8 

9 

32 

10 

20 

30 

40 

49 

59 

69 

79 

89 

99 

I 

2 

4 

5 

6 

7 

8 

9 

33 

10 

20 

31 

4i 

5i 

61 

71 

81 

92 

102 

I 

2 

4 

5 

6 

7 

8 

9 

34 

10 

21 

31 

42 

52 

63 

73 

84 

94 

105 

I 

2 

4 

c 

6 

8 

9 

10 

35 

ii 

22 

32 

43 

54 

65 

76 

86 

97 

108 

I 

2 

4 

6 

7 

8 

9 

10 

36 

ii 

22 

33 

44 

56 

67 

78 

89 

100 

in 

I 

2 

6 

7 

8 

9 

10 

37 

ii 

23 

34 

46 

57 

69 

80 

9i 

103 

114 

I 

2 

3 

6 

7 

8 

9 

ii 

38 

12 

23 

35 

47 

59 

70 

82 

94 

106 

117 

I 

2 

4 

6 

7 

8 

10 

ii 

39 

12 

1L 

36 

48 

60 

72 

84 

96 

108 

1  20 

I 

4 

6 

7 

9 

10 

ii 

40 

12 

25 

37 

49 

62 

74 

86 

99 

in 

123 

I 

4 

6 

8 

9 

10 

ii 

4i 

13 

25 

38 

5i 

63 

76 

89 

IOI 

114 

127 

I 

4 

6 

8 

9 

10 

2 

42 

J 

26 

39 

52 

65 

78 

91 

IO/ 

117 

130 

I 

4 

7 

8 

9 

ii 

2 

43 

fl 

27 

40 

53 

66 

80 

93 

106 

119 

133 

I 

4 

7 

8 

10 

n 

2 

44 

L 

27 

41 

54 

68 

81 

95 

109 

122 

136 

I 

4 

6 

7 

8 

10 

ii 

' 

45 

L 

28 

42 

56 

69 

83 

97 

in 

!25 

139 

I 

4 

6 

7 

9 

10 

ii 

, 

46 

i 

28 

43 

57 

7J 

85 

99 

ii/ 

128 

142 

I 

4 

6 

7 

9 

10 

12 

\ 

47 

I 

29 

44 

58 

73 

87 

102 

116 

131 

H5 

I 

4 

6 

7 

9 

10 

12 

\ 

48 

: 

3° 

44 

59 

74 

89 

IO/ 

119 

133 

148 

r- 

£ 

6 

8 

9 

n 

12 

Iz 

49 

( 

3° 

45 

60 

76 

9i 

1  06 

121 

136 

151 

2 

c 
w 

6 

8 

9 

ii 

12 

\i 

50 

I 

3i 

46 

62 

77 

93 

108 

123 

139 

154 

.7 

.^ 

•J 

•4 

•5 

.6 

•7 

.8 

•9 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

O.I 

0.2 

i 

] 

0.3 

i 

I 

i 

I 

Correction  for  tenths 

0.4 

i 

i 

I 

i 

I 

0-5 

i 

i 

I 

i 

2 

of  width7—  width. 

0.6 

i 

i 

i 

I 

2 

2 

0.7 

i 

i 

2 

2 

2 

2 

0.8 

] 

i 

2 

2 

2 

2 

0.9 

i 

i 

i 

2 

2 

2 

2 

(14) 


CORRECTIONS. 


[w/_w  j_50:  h— hx  11-20.] 


.t 

.2 

•3 

-4 

•5 

.6 

•  7 

.8 

•9 

w'  —  w. 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

I 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

i 

2 

7 

7 

8 

9 

9 

10 

10 

ii 

12 

12 

i 

i 

i 

3 

10 

ii 

12 

13 

14 

15 

16 

18 

19 

i 

i 

i 

i 

4 

14 

15 

16 

17 

19 

20 

21 

22 

23 

-7 

25 

i 

1 

i 

i 

i 

5 

17 

19 

20 

22 

23 

25 

26 

28 

29 

31 

i 
i 

i 

i 

i 

2 

2 

2 
2 

6 

7 

20 
24 

22 
26 

24 

28 

26 

30 

28 
32 

30 

35 

31 

37 

33 
39 

35 

37 
43 

i 

2 

2 

2 

8 

27 

30 

32 

35 

37 

40 

42 

44 

47 

i 

i 

2 

2 

2 

2 

9 

31 

33 

36 

39 

42 

44 

47 

50 

53 

56 

i 

2 

2 

2 

2 

3 

10 

34 

37 

40 

43 

46 

49 

52 

56 

59 

62 

2 

2 

2 

3 

3 

ii 

37 

41 

44 

48 

5I 

54 

58 

61 

65 

68 

2 

2 

3 

3 

3 

12 

41 

44 

48 

52 

56 

59 

63 

67 

70 

74 

2 

2 

2 

3 

3 

4 

13 

44 

48 

52 

56 

60 

64 

68 

72 

80 

2 

2 

3 

3 

3 

4 

14 

48 

52 

56 

60 

65 

69 

73 

78 

82 

86 

2 

2 

3 

3 

4 

4 

15 

51 

56 

60 

65 

69 

74 

79 

83 

88 

93 

i 

2 

3 

3 

3 

4 

4 

16 

54 

59 

64 

69 

74 

79 

84 

89 

94 

99 

2 

2 

3 

3 

4 

4 

5 

17 

58 

63 

68 

73 

79 

84 

89 

94 

100 

105 

2 

2 

3 

3 

4 

4 

5 

18 

61 

67 

72 

78 

83 

89 

94 

100 

106 

in 

2 

2 

3 

4 

4 

5 

5 

19 

65 

7o 

76 

82 

88 

94 

100 

106 

in 

117 

2 

2 

3 

4 

4 

5 

6 

20 

68 

74 

80 

86 

93 

99 

105 

in 

117 

123 

2 

3 

3 

4 

5 

5 

6 

21 

71 

78 

84 

9I 

97 

104 

no 

117 

123 

130 

2 

3 

3 

4 

5 

5 

6 

22 

75 

81 

88 

95 

102 

109 

115 

122 

129 

136 

2 

3 

4 

4 

5 

6 

6 

23 

78 

85 

92 

99 

106 

114 

121 

128 

142 

2 

3 

4 

4 

5 

6 

7 

24 

81 

89 

96 

104 

in 

119 

126 

T33 

141 

148 

2 

2 

3 

4 

5 

5 

6 

7 

25 

85 

93 

loo 

108 

116 

123 

J31 

139 

147 

154 

I 

2 

2 

3 

4 

5 

6 

6 

7 

26 

88 

96 

104 

112 

120 

128 

136 

144 

152 

160 

I 

2 

3 

3 

4 

5 

6 

7 

8 

27 

92 

TOO 

108 

117 

125 

133 

142 

150 

158 

167 

I 

2 

3 

3 

4 

5 

6 

7 

8 

28 

95 

104 

112 

121 

130 

138 

147 

156 

164 

173 

I 

2 

3 

4 

4 

5 

6 

7 

8 

29 

98 

107 

116 

125 

134 

143 

152 

161 

170 

179 

I 

2 

3 

4 

5 

6 

6 

7 

8 

30 

102 

III 

1  20 

130 

139 

148 

157 

167 

176 

185 

I 

2 

3 

4 

5 

6 

7 

8 

9 

31 

105 

U5 

124 

134 

144 

153 

163 

172 

182 

191 

I 

2 

3 

4 

5 

6 

7 

8 

9 

32 

I09 

119 

128 

138 

148 

•158 

168 

178 

188 

198 

I 

2 

3 

4 

5 

6 

7 

8 

9 

33 

112 

122 

132 

143 

153 

163 

173 

183 

194 

204 

I 

2 

3 

4 

5 

6 

7 

8 

9 

34 

115 

126 

136 

147 

157 

168 

178 

189 

199 

210 

I 

2 

3 

4 

5 

6 

8 

9 

10 

35 

II9 

130 

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199 

203 

207 

212 

216 

2 

2 

3 

3 

4 

4 

15 

190 

194 

199 

2O4 

208 

213 

218 

222 

227 

231 

i 

2 

3 

3 

3 

4 

4 

16 

202 

207 

212 

217 

222 

227 

232 

237 

242 

247 

2 

2 

3 

3 

4 

4 

5 

17 

2I5 

220 

226 

23I 

236 

241 

247 

252 

257 

262 

2 

2 

3 

3 

4 

4 

5 

18 

228 

233 

239 

244 

250 

256 

261 

267 

272 

278 

2 

2 

3 

4 

4 

5 

5 

19 

240 

246 

252 

258 

264 

270 

276 

28l 

287 

293 

2 

2 

3 

4 

4 

5 

6 

20 

253 

259 

265 

272 

278 

284 

290 

296 

302 

309 

2 

3 

3 

4 

5 

5 

6 

21 

266 

272 

279 

285 

292 

298 

305 

311 

318 

324 

2 

3 

3 

4 

5 

5 

6 

22 

278 

285 

292 

299 

306 

312 

319 

326 

333 

340 

2 

3 

4 

4 

5 

6 

6 

23 

291 

298 

3°5 

312 

319 

327 

334 

341 

348 

355 

I 

2 

3 

4 

4 

5 

6 

7 

24 

304 

311 

3!9 

326 

333 

34i 

348 

356 

363 

37o 

2 

2 

3 

4 

5 

5 

6 

7 

25 

316 

324 

332 

340 

347 

355 

363 

370 

378 

386 

2 

2 

3 

4 

5 

6 

6 

7 

26 

329 

337 

345 

353 

36i 

369 

377 

385 

393 

401 

2 

3 

3 

4 

5 

6 

7 

8 

27 

342 

350 

358 

367 

375 

383 

392 

400 

408 

417 

2 

3 

3 

4 

5 

6 

7 

8 

28 

354 

363 

372 

380 

389 

398 

406 

415 

423 

432 

2 

3 

4 

4 

5 

6 

7 

8 

29 

367 

376 

385 

394 

403 

412 

421 

430 

439 

448 

2 

3 

4 

5 

6 

6 

7 

8 

30 

380 

389 

398 

407 

417 

426 

435 

444 

454 

463 

2 

3 

4 

5 

6 

7 

8 

9 

31 

392 

402 

411 

421 

43i 

440 

450 

459 

469 

478 

2 

3 

4 

5 

6 

7 

8 

9 

32 

405 

415 

425 

435 

444 

454 

464 

474 

484 

494 

2 

3 

4 

5 

6 

7 

8 

9 

33 

418 

428 

438 

448 

458 

469 

479 

489 

499 

509 

2 

3 

4 

5 

6 

7 

8 

9 

34 

430 

441 

45i 

462 

472 

483 

493 

504 

5H 

525 

2 

3 

4 

5 

6 

8 

9 

10 

35 

443 

454 

465 

475 

486 

497 

508 

519 

529 

540 

2 

3 

4 

6 

7 

8 

9 

10 

36 

456 

467 

478 

489 

500 

5ii 

522 

533 

544 

556 

2 

3 

5 

6 

7 

8 

9 

10 

37 

468 

480 

491 

502 

5H 

525 

537 

548 

560 

57i 

I 

2 

4 

5 

6 

7 

8 

9 

II 

38 

481 

493 

504 

5i6 

528 

540 

55i 

563 

575 

586 

I 

2 

4 

5 

6 

7 

8 

10 

II 

39 

494 

506 

5i8 

530 

542 

554 

566 

578 

590 

602 

I 

2 

4 

5 

6 

7 

9 

10 

II 

40 

506 

5i9 

53i 

543 

556 

568 

580 

593 

605 

617 

3 

4 

5 

6 

8 

9 

10 

II 

4i 

519 

531 

544 

557 

569 

582 

595 

607 

620 

633 

3 

4 

5 

6 

8 

9 

10 

12 

42 

531 

544 

557 

570 

583 

596 

609 

622 

635 

648 

3 

4 

5 

7 

8 

9 

II 

12 

43 

544 

557 

57i 

584 

597 

610 

624 

637 

650 

664 

3 

4 

5 

7 

8 

10 

II 

12 

44 

557 

570 

584 

598 

611 

625 

638 

652 

665 

679 

3 

4 

6 

7 

8 

10 

II 

13 

45 

569 

583 

597 

611 

625 

639 

653 

667 

681 

694 

I 

3 

4 

6 

7 

9 

10 

II 

T3 

46 

582 

596 

610 

625 

639 

653 

667 

68  1 

696 

710 

I 

3 

4 

6 

7 

9 

10 

12 

13 

47 

595 

609 

624 

638 

653 

667 

682 

696 

711 

725 

I 

3 

4 

6 

9 

10 

12 

13 

48 

607 

622 

637 

652 

667 

681 

696 

711 

726 

741 

2 

3 

5 

6 

8 

9 

II 

12 

14 

49 

620 

635 

650 

665 

68  1 

696 

711 

726 

74i 

756 

2 

3 

5 

6 

8 

9 

II 

12 

14 

50 

633 

648 

664 

679 

694 

710 

725 

741 

756 

772 

./ 

.2 

v? 

-4 

•5 

.6 

•7 

.8 

-9 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

O.I 

i 

i 

i 

i 

i 

i 

i 

i 

2 

2 

0.2 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

0.3 

4 

4 

4 

4 

4 

4 

4 

4 

5 

5 

Correction  for  tenths 

0.4 

5 

5 

5 

5 

6 

6 

6 

6 

6 

6 

°-5 

6 

6 

7 

7 

7 

7 

7 

7 

8 

8 

of  width'—  width. 

0.6 

8 

8 

8 

8 

8 

9 

9 

9 

9 

9 

0.7 

9 

9 

9 

10 

10 

10 

10 

10 

ii 

ii 

0.8 

10 

10 

ii 

ii 

ii 

ii 

12 

12 

12 

12 

0.9 

ii 

12 

12 

12 

T3 

13 

13 

13 

14 

14 

(18) 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


DEC  1  4  HI 

SEP  1  o  ' 


LIBRARY,    BRANCH   OP   THE   COLLEGE   OP  AGRICULTURE 

5m-9,'35(s) 


22769 
Crandall 
Tables 


fcr 


and  other  e 

Q  1  4  '41 

SEP  1  Q  '56 


•    L-. 
the 


OT222 

12. 

1893 


TFZZZ 

C7 

1833 


LIBRARY,  BRANCH  OF  THE  COLLEGE  OF  AGRICULTURE,  DAVIS 
UNIVERSITY  OF  CALIFORNIA 


